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a(n) is the number of 2-regular 4-hypergraphs on 2n labeled vertices. (In a r-hypergraph, each hyper-edge is a proper r-set; k-regular implies that each vertex is in exactly k hyperedges.)
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%I #11 Dec 25 2017 03:50:07

%S 1,0,0,15,1855,469980,214402650,160081596675,182667234224475,

%T 302414315250247200,697372026302486234700,2167773244010692751057625,

%U 8842276105055583472501844625,46275602006744820263447546152500

%N a(n) is the number of 2-regular 4-hypergraphs on 2n labeled vertices. (In a r-hypergraph, each hyper-edge is a proper r-set; k-regular implies that each vertex is in exactly k hyperedges.)

%C P-recursive.

%H Marni Mishna, <a href="https://web.archive.org/web/20070623105521/http://algo.inria.fr/mishna/Reg-Asympt/reg-asympt.html">Maple worksheet</a>

%F Differential equation satisfied by exponential generating function sum a(n) t^(2n)/(2n)! {F(0) = 1, -144*t^3*(-2 + t^2)^2*(d^2/dt^2)F(t) - 12*(-2 + t^2)*(2*t^8-t^6 + 72 + 6*t^4-108*t^2)*(d/dt)F(t) - t^5*(-2 + t^2)*(t^2-3)*(t^4 + 4*t^2 + 36)*F(t)}.

%F Linear recurrence for a(n): {(15067980*n + 10550232*n^6 + 2859384*n^7 + 522720*n^8 + 128*n^11 + 2494800 + 61600*n^9 + 4224*n^10 + 52629038*n^3 + 45995730*n^4 + 26679070*n^5 + 37729494*n^2)*a(n) + (3791790*n + 109368*n^6 + 13872*n^7 + 1008*n^8 + 1247400 + 32*n^9 + 3747208*n^3 + 1767087*n^4 + 543858*n^5 + 4994577*n^2)*a(n + 1) + (28354500*n + 154560*n^6 + 11712*n^7 + 384*n^8 + 15478428*n^3 + 5309976*n^4 + 1152480*n^5 + 27874680*n^2 + 12474000)*a(n + 2) + (-623700-794025*n-48*n^6-115380*n^3-17760*n^4-1440*n^5-416757*n^2)*a(n + 3) + (599130*n + 534600 + 267282*n^2 + 59328*n^3 + 6552*n^4 + 288*n^5)*a(n + 4) + (-14166*n-26730-2484*n^2-144*n^3)*a(n + 5) + 54*a(n + 6), a(3) = 15, a(4) = 1855, a(5) = 469980, a(0) = 1, a(1) = 0, a(2) = 0}

%F Recurrence (of order 5): 54*(3*n - 4)*a(n) = 18*(n-1)*(2*n - 1)*(12*n^2 - 16*n + 9)*a(n-1) + 18*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*(3*n + 1)*a(n-2) + 3*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(24*n^2 - 65*n + 24)*a(n-3) + 3*(n-3)*(n-2)*(n-1)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(3*n + 1)*a(n-4) + 2*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(3*n - 1)*a(n-5). - _Vaclav Kotesovec_, Mar 11 2014

%F a(n) ~ 2^(3*n+1) * n^(3*n) / (3^n * exp(3*n+3/2)). - _Vaclav Kotesovec_, Mar 11 2014

%e One of the 15 2-regular 4-hypergraphs on 6 vertices: {{1234},{4561}, {2356}}.

%Y Cf. A025035, A110100, A110101, A025036.

%K easy,nonn

%O 0,4

%A _Marni Mishna_, Jul 11 2005

%E Replaced broken link, _Vaclav Kotesovec_, Mar 11 2014